Semi-global solutions to the Goursat problem for second-order hyper-quasilinear hyperbolic systems with lineary dependent principal coefficients and applications to the vacuum Einstein equations

TL;DR

This paper extends the Goursat problem solutions for second-order hyperbolic systems with linearly dependent coefficients, applying the results to semi-global existence in vacuum Einstein equations.

Contribution

It broadens previous results to hyper-quasilinear systems with linearly dependent coefficients, providing semi-global solutions and applications to Einstein's vacuum equations.

Findings

Solutions exist near meeting characteristic hypersurfaces.

Established semi-global existence and uniqueness for vacuum Einstein equations.

Extended prior work to more general hyperbolic systems.

Abstract

In this work, we significantly extend the results of D. Houpa, 2006 on the Goursat problem for second-order semi-linear hyperbolic systems to the broader framwork of second-order hyper-quasilinear hyperbolic systems of Goursat type, in which the coefficients of the second-order derivatives depend linearly on the unknown. By adapting techniques inspired by Y. Foures (Choquet)- Bruhat, Acta Mathematica, 1952. we show that in the Sobolev type spaces for the Goursat problem quasilinear hyperbolic of the second order considered, the solution exists and is defined in the vicinity of the meeting characteristic hypersurfaces which carry the initial data. As an application, in harmonic gauge, we derive a semi-global existence and uniqueness result for the vacuum Einstein equations.